You can control the contrast in lith prints by adjusting the exposure and development time. Even when the paper's fogged you can still get strong contrast by under-exposing the paper (for a lith print, i.e. over-expose one or two stops compared with a normal print - just the act of over-exposing can also get round some of the issues with fogged paper) and then developing until you get the shadow density that you desire. Also, lith prints can be very attractive without the need to be 100% accurate to the original negative so fogging can be less obvious.